Question

A) Write the polynomial expression in simplest form:

2x-10/x-25 * x^2+6x+5/x^2+10x+21

B) What are the restrictions on the variable?

MUST SHOW WORK

Answer 1

To simplify the polynomial expression and identify any restrictions, we will follow these steps:

### Part A: Simplifying the Expression

The expression is given as:

\[
\frac{2x - 10}{x - 25} * \frac{x^2 + 6x + 5}{x^2 + 10x + 21}
\]

**Step 1: Factor each polynomial**

1. **Factor \(2x - 10\):**
\[
2x - 10 = 2(x - 5)
\]

2. **Factor \(x^2 + 6x + 5\):**
This can be factored by finding two numbers that multiply to \(5\) and add up to \(6\): which are \(1\) and \(5\).
\[
x^2 + 6x + 5 = (x + 1)(x + 5)
\]

3. **Factor \(x^2 + 10x + 21\):**
This can be factored by finding two numbers that multiply to \(21\) and add up to \(10\): which are \(3\) and \(7\).
\[
x^2 + 10x + 21 = (x + 3)(x + 7)
\]

**Step 2: Rewrite the expression with the factors**

Now substituting back into our expression:

\[
\frac{2(x - 5)}{x - 25} * \frac{(x + 1)(x + 5)}{(x + 3)(x + 7)}
\]

**Step 3: Simplifying the expression**

Now, we can combine the fractions:

\[
\frac{2(x - 5)(x + 1)(x + 5)}{(x - 25)(x + 3)(x + 7)}
\]

At this point, we notice that \(x - 25\) does not factor into anything related to our other polynomials, so we cannot cancel any terms.

The expression is thus:

\[
\frac{2(x - 5)(x + 1)(x + 5)}{(x - 25)(x + 3)(x + 7)}
\]

### Conclusion for Part A

The polynomial expression in simplest form is:

\[
\frac{2(x - 5)(x + 1)(x + 5)}{(x - 25)(x + 3)(x + 7)}
\]

### Part B: Finding Restrictions on the Variable

**Restrictions occur when the denominator is zero.** Thus, we need to find the values of \(x\) for which the denominator equals zero:

1. From \(x - 25 = 0\):
\[
x = 25
\]

2. From \(x + 3 = 0\):
\[
x = -3
\]

3. From \(x + 7 = 0\):
\[
x = -7
\]

### Conclusion for Part B

The restrictions on the variable \(x\) are:

\[
x \neq 25, x \neq -3, x \neq -7
\]